1 a positive definite matrix 1 one of n i i i n i i i

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1 , a positive definite matrix. 1 , one of n i i i n i i i E n E n - - = = - → - = H H g g 0 θ θ θ θ ( 29 the regularity conditions. Therefore, collecting terms, ˆ ˆ or plim = MLE MLE - 0 θ θ θ θ
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Part 18: Maximum Likelihood Estimation Asymptotic Variance ™  40/47
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Part 18: Maximum Likelihood Estimation Asymptotic Variance ™  41/47
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Part 18: Maximum Likelihood Estimation ™  42/47
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Part 18: Maximum Likelihood Estimation ™  43/47
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Part 18: Maximum Likelihood Estimation Asymptotic Distribution ™  44/47
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Part 18: Maximum Likelihood Estimation Efficiency: Variance Bound ™  45/47
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Part 18: Maximum Likelihood Estimation Invariance The maximum likelihood estimator of a function of , say h( ) is h(MLE). This is not always true of other kinds of estimators. To get the variance of this function, we would use the delta method. E.g., the MLE of θ =( β /σ) is b /( ee /n) ™  46/47
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Part 18: Maximum Likelihood Estimation Estimating the Tobit Model   47/47 σ - - Φ + φ ÷ ÷ ÷ σ σ σ = i i β x x β β n i i i i= 1 i i i Log likelihood for the tobit model for estimation of   and  : y 1 logL= (1-d ) log d log d 1 if y 0,  0 if y  =  0.  Derivatives are very complicated, Hessian is  ( 29 ( 29 ( 29 ( 29 ( 29 θ σ σ σ θ θ Φ + θφ θ + Φ + θ + π - θ + i i i i = - β β x x x x n i i i i= 1 2 i i i nightmarish.  Consider the Olsen transformation* : = 1/ ,  = - / . (One to one;  =1 / / logL= log (1-d ) log d log y          log (1-d ) log d (log (1 / 2) log2 (1 / 2) y ) γ γ .29 γ γ γ γ ( 29 ( 29 = = φ = - Φ = - ÷ ∂θ θ i i i x x x n i= 1 n i i i i 1 n i i i i 1 logL (1-d ) de logL 1 d e y *Note on the Uniqueness of the MLE in the Tobit Model," Econometrica, 1978. γ γ γ
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